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Fix the two unrendered ones as well.
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@kellertuer
kellertuer committed Nov 16, 2023
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157 changes: 79 additions & 78 deletions docs/src/tutorials/GeodesicRegression.md
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Expand Up @@ -115,7 +115,7 @@ end
```

For the Euclidean case, the result is given by the first principal component of a principal component analysis,
see [PCR](https://en.wikipedia.org/wiki/Principal_component_regression), which is given by $p^* = \frac{1}{n}\displaystyle\sum_{i=1}^n d_i$ and
see [PCR](https://en.wikipedia.org/wiki/Principal_component_regression) which is given by $p^* = \frac{1}{n}\displaystyle\sum_{i=1}^n d_i$ and
the direction $X^*$ is obtained by defining the zero mean data matrix

``` math
Expand Down Expand Up @@ -177,11 +177,11 @@ y = gradient_descent(
)
```

Initial | F(x): 0.142862
# 50 | F(x): 0.141113
# 100 | F(x): 0.141113
# 150 | F(x): 0.141113
# 200 | F(x): 0.141113
Initial | f(x): 0.142862
# 50 | f(x): 0.141113
# 100 | f(x): 0.141113
# 150 | f(x): 0.141113
# 200 | f(x): 0.141113
The algorithm reached its maximal number of iterations (200).

([0.7119768725361988, 0.009463059143003981, 0.7021391482357537], [0.590008151835008, -0.5543272518659472, -0.5908038715512287])
Expand Down Expand Up @@ -266,73 +266,73 @@ y2 = gradient_descent(
);
```

Initial | F(x): 0.089844
# 3 | F(x): 0.085364
# 6 | F(x): 0.085364
# 9 | F(x): 0.085364
# 12 | F(x): 0.085364
# 15 | F(x): 0.085364
# 18 | F(x): 0.085364
# 21 | F(x): 0.085364
# 24 | F(x): 0.085364
# 27 | F(x): 0.085364
# 30 | F(x): 0.085364
# 33 | F(x): 0.085364
# 36 | F(x): 0.085364
# 39 | F(x): 0.085364
# 42 | F(x): 0.085364
# 45 | F(x): 0.085364
# 48 | F(x): 0.085364
# 51 | F(x): 0.085364
# 54 | F(x): 0.085364
# 57 | F(x): 0.085364
# 60 | F(x): 0.085364
# 63 | F(x): 0.085364
# 66 | F(x): 0.085364
# 69 | F(x): 0.085364
# 72 | F(x): 0.085364
# 75 | F(x): 0.085364
# 78 | F(x): 0.085364
# 81 | F(x): 0.085364
# 84 | F(x): 0.085364
# 87 | F(x): 0.085364
# 90 | F(x): 0.085364
# 93 | F(x): 0.085364
# 96 | F(x): 0.085364
# 99 | F(x): 0.085364
# 102 | F(x): 0.085364
# 105 | F(x): 0.085364
# 108 | F(x): 0.085364
# 111 | F(x): 0.085364
# 114 | F(x): 0.085364
# 117 | F(x): 0.085364
# 120 | F(x): 0.085364
# 123 | F(x): 0.085364
# 126 | F(x): 0.085364
# 129 | F(x): 0.085364
# 132 | F(x): 0.085364
# 135 | F(x): 0.085364
# 138 | F(x): 0.085364
# 141 | F(x): 0.085364
# 144 | F(x): 0.085364
# 147 | F(x): 0.085364
# 150 | F(x): 0.085364
# 153 | F(x): 0.085364
# 156 | F(x): 0.085364
# 159 | F(x): 0.085364
# 162 | F(x): 0.085364
# 165 | F(x): 0.085364
# 168 | F(x): 0.085364
# 171 | F(x): 0.085364
# 174 | F(x): 0.085364
# 177 | F(x): 0.085364
# 180 | F(x): 0.085364
# 183 | F(x): 0.085364
# 186 | F(x): 0.085364
# 189 | F(x): 0.085364
# 192 | F(x): 0.085364
# 195 | F(x): 0.085364
# 198 | F(x): 0.085364
Initial | f(x): 0.089844
# 3 | f(x): 0.085364
# 6 | f(x): 0.085364
# 9 | f(x): 0.085364
# 12 | f(x): 0.085364
# 15 | f(x): 0.085364
# 18 | f(x): 0.085364
# 21 | f(x): 0.085364
# 24 | f(x): 0.085364
# 27 | f(x): 0.085364
# 30 | f(x): 0.085364
# 33 | f(x): 0.085364
# 36 | f(x): 0.085364
# 39 | f(x): 0.085364
# 42 | f(x): 0.085364
# 45 | f(x): 0.085364
# 48 | f(x): 0.085364
# 51 | f(x): 0.085364
# 54 | f(x): 0.085364
# 57 | f(x): 0.085364
# 60 | f(x): 0.085364
# 63 | f(x): 0.085364
# 66 | f(x): 0.085364
# 69 | f(x): 0.085364
# 72 | f(x): 0.085364
# 75 | f(x): 0.085364
# 78 | f(x): 0.085364
# 81 | f(x): 0.085364
# 84 | f(x): 0.085364
# 87 | f(x): 0.085364
# 90 | f(x): 0.085364
# 93 | f(x): 0.085364
# 96 | f(x): 0.085364
# 99 | f(x): 0.085364
# 102 | f(x): 0.085364
# 105 | f(x): 0.085364
# 108 | f(x): 0.085364
# 111 | f(x): 0.085364
# 114 | f(x): 0.085364
# 117 | f(x): 0.085364
# 120 | f(x): 0.085364
# 123 | f(x): 0.085364
# 126 | f(x): 0.085364
# 129 | f(x): 0.085364
# 132 | f(x): 0.085364
# 135 | f(x): 0.085364
# 138 | f(x): 0.085364
# 141 | f(x): 0.085364
# 144 | f(x): 0.085364
# 147 | f(x): 0.085364
# 150 | f(x): 0.085364
# 153 | f(x): 0.085364
# 156 | f(x): 0.085364
# 159 | f(x): 0.085364
# 162 | f(x): 0.085364
# 165 | f(x): 0.085364
# 168 | f(x): 0.085364
# 171 | f(x): 0.085364
# 174 | f(x): 0.085364
# 177 | f(x): 0.085364
# 180 | f(x): 0.085364
# 183 | f(x): 0.085364
# 186 | f(x): 0.085364
# 189 | f(x): 0.085364
# 192 | f(x): 0.085364
# 195 | f(x): 0.085364
# 198 | f(x): 0.085364
The algorithm reached its maximal number of iterations (200).

For plotting we again generate all data
Expand All @@ -350,7 +350,7 @@ geo_conn_highlighted2 = shortest_geodesic(

## Unlabeled Data

If we are not given time points $t_i$, then the optimization problem extends informally speaking,
If we are not given time points $t_i$, then the optimization problem extends, informally speaking,
to also finding the “best fitting” (in the sense of smallest error).
To formalize, the objective function here reads

Expand All @@ -373,7 +373,7 @@ N = M × Euclidean(length(t2))

ProductManifold with 2 submanifolds:
TangentBundle(Sphere(2, ℝ))
Euclidean(7; field = ℝ)
Euclidean(7; field=ℝ)

``` math
\operatorname*{arg\,min}_{\bigl((p,X),t\bigr)\in\mathcal N} F(p, X, t).
Expand Down Expand Up @@ -485,9 +485,9 @@ y3 = alternating_gradient_descent(
)
```

Initial | F(x): 0.089844
# 50 | F(x): 0.091097
# 100 | F(x): 0.091097
Initial | f(x): 0.089844
# 50 | f(x): 0.091097
# 100 | f(x): 0.091097
The algorithm reached its maximal number of iterations (100).

(ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}(([0.750222090700214, 0.031464227399200885, 0.6604368380243274], [0.6636489079535082, -0.3497538263293046, -0.737208025444054])), [0.7965909273713889, 0.43402264218923514, 0.755822122896529, 0.001059348203453764, -0.6421135044471217, -0.8635572995105818, -0.5546338813212247])
Expand Down Expand Up @@ -517,3 +517,4 @@ Note that the geodesics from the data to the regression geodesic meet at a nearl
Pages = ["GeodesicRegression.md"]
Canonical=false
```

2 changes: 1 addition & 1 deletion docs/src/tutorials/InplaceGradient.md
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Expand Up @@ -5,7 +5,7 @@ When it comes to time critical operations, a main ingredient in Julia is given b
mutating functions, that is those that compute in place without additional memory
allocations. In the following, we illustrate how to do this with `Manopt.jl`.

Let’s start with the same function as in [🏔️ Get started: optimize.](https://manoptjl.org/stable/tutorials/Optimize.html)
Let’s start with the same function as in [Get Started: Optimize!](https://manoptjl.org/stable/tutorials/Optimize!.html)
and compute the mean of some points, only that here we use the sphere $\mathbb S^{30}$
and $n=800$ points.

Expand Down
8 changes: 4 additions & 4 deletions tutorials/GeodesicRegression.qmd
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Expand Up @@ -77,7 +77,7 @@ render_asymptote(img_folder * "/regression_data.asy"; render=render_size);
## Time Labeled Data

If for each data item $d_i$ we are also given a time point $t_i\in\mathbb R$, which are pairwise different,
then we can use the least squares error to state the objetive function as [Fletcher:2013](@cite)
then we can use the least squares error to state the objective function as [Fletcher:2013](@cite)

```math
F(p,X) = \frac{1}{2}\sum_{i=1}^n d_{\mathcal M}^2(γ_{p,X}(t_i), d_i),
Expand Down Expand Up @@ -349,7 +349,7 @@ where $t = (t_1,\ldots,t_n) \in \mathbb R^n$ is now an additional parameter of t
We write $F_1(p, X)$ to refer to the function on the tangent bundle for fixed values of $t$ (as the one in the last part)
and $F_2(t)$ for the function $F(p, X, t)$ as a function in $t$ with fixed values $(p, X)$.

For the Euclidean case, there is no neccessity to optimize with respect to $t$, as we saw
For the Euclidean case, there is no necessity to optimize with respect to $t$, as we saw
above for the initialization of the fixed time points.

On a Riemannian manifold this can be stated as a problem on the product manifold $\mathcal N = \mathrm{T}\mathcal M \times \mathbb R^n$, i.e.
Expand All @@ -363,7 +363,7 @@ N = M × Euclidean(length(t2))
```

In this tutorial we present an approach to solve this using an alternating gradient descent scheme.
To be precise, we define the cost funcion now on the product manifold
To be precise, we define the cost function now on the product manifold

```{julia}
struct RegressionCost2{T}
Expand Down Expand Up @@ -414,7 +414,7 @@ function (a::RegressionGradient2a!)(N, Y, x)
end
```

Finally, we addionally look for a fixed point $x=(p,X) ∈ \mathrm{T}\mathcal M$ at
Finally, we additionally look for a fixed point $x=(p,X) ∈ \mathrm{T}\mathcal M$ at
the gradient with respect to $t∈\mathbb R^n$, the second component, which is given by

```math
Expand Down
2 changes: 1 addition & 1 deletion tutorials/InplaceGradient.qmd
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Expand Up @@ -3,7 +3,7 @@ title: "Speedup using in-place evaluation"
author: Ronny Bergmann
---

When it comes to time critital operations, a main ingredient in Julia is given by
When it comes to time critical operations, a main ingredient in Julia is given by
mutating functions, that is those that compute in place without additional memory
allocations. In the following, we illustrate how to do this with `Manopt.jl`.

Expand Down

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